document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=437478>Question 437478</A>: <I><SPAN STYLE=\"word-break: break-all;\"> TriangleABC is an acute triangle. Two of its side measure 11cm and 14cm. What is the rangle of possible values for its third side? </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #302682 by </B><A HREF=/tutors/aboutme.mpl?userid=richard1234><B>richard1234(7193)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=richard1234><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=richard1234><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=302682\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> A corollary from the law of cosines, a triangle with sides a,b,c is acute if and only if <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=a%5E2+%2B+b%5E2+%3E+c%5E2 ALIGN=MIDDLE  ALT=\"a%5E2+%2B+b%5E2+%3E+c%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > (this inequality is also symmetric, so it applies if you switch the ordering of a,b,c). Hence, for a triangle with sides 11, 14, and x, all three inequalities must be satisfied:\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%5E2+%2B+14%5E2+%3E+x%5E2 ALIGN=MIDDLE  ALT=\"11%5E2+%2B+14%5E2+%3E+x%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3C+sqrt%28317%29 ALIGN=MIDDLE  ALT=\"x+%3C+sqrt%28317%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=11%5E2+%2B+x%5E2+%3E+14%5E2 ALIGN=MIDDLE  ALT=\"11%5E2+%2B+x%5E2+%3E+14%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=x+%3E+sqrt%2875%29+=+5sqrt%283%29 ALIGN=MIDDLE  ALT=\"x+%3E+sqrt%2875%29+=+5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=14%5E2+%2B+x%5E2+%3E+11%5E2 ALIGN=MIDDLE  ALT=\"14%5E2+%2B+x%5E2+%3E+11%5E2\"   DISABLED_style= \"max-width:90%; height:auto;\" > --> all values x satisfy this inequality.\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( "From the first inequality, the solution set for x is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=-infinity ALIGN=MIDDLE  ALT=\"-infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28317%29 ALIGN=MIDDLE  ALT=\"sqrt%28317%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >) and from the second inequality, the solution set is (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5sqrt%283%29 ALIGN=MIDDLE  ALT=\"5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=infinity ALIGN=MIDDLE  ALT=\"infinity\"   DISABLED_style= \"max-width:90%; height:auto;\" >). Taking the intersection of these sets, the only possible values for x are those lying in the set (<IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=5sqrt%283%29 ALIGN=MIDDLE  ALT=\"5sqrt%283%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >, <IMG STYLE=\"max-width:80%;\" SRC=/cgi-bin/plot-formula.mpl?expression=sqrt%28317%29 ALIGN=MIDDLE  ALT=\"sqrt%28317%29\"   DISABLED_style= \"max-width:90%; height:auto;\" >).\r<BR>\n" );
document.write( "<BR>\n" );
document.write( "\n" );
document.write( " </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );